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Introduction to Radicals - Problem 4
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Simplifying the square root of a fraction. So simplifying the square root of a fraction is very similar to simplifying the square root of a product. Basically what we have to think about is that this square root is going to everything inside of it. So if we’re multiplying it can get split up into the square root of two things times each other. With division we can think of it the exact same way except the square root of the two things divided by each other.

For this particular problem we can break it up square root of 36 over the square root of 49. Square root of 36 is 6, square root of 49 is 7. So we’re just left with 6/7.

Similar idea for cube root. We could break this up, I’m going to skip the step of actually writing it up this time but basically what we’re looking at is the cube root of -8 over the cube root of 125. Cube root of -8 is just -2, cube root of 125 is 5.

Moving down the row, so here we’re dealing with the square root of another quotient, we could split it up into the square root of the numerator over square root of the denominator so the square root of 3 over the square root of 25. Square root of 25 we know is 5 but for this one, square root of 3 doesn’t have a perfect square so we just leave it as the square root of 3. It’s not the nicest number in the world but it’s the most we can do with this.

The last one is a 4th root. The principle still stays the same but now we’re figuring out we need something to go in 4 times. 81 is 9², 9 is 3² so this actually is 3 to the 4th. So the 4th root of 81 is 3 and now we need to figure out what the 4th root of x to the 9th is. The 4th root of x to the 4th is x, and we actually have two x to the 4th so this turns out to be x². That’ll take care of either of those xs in the inside but we still have that one left over. So we’re just left with times the 4th root of x.

So simplifying a quotient, very, very similar to simplifying a product just that we have a fraction.

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